Inverse distance weighting

From Wikipedia, the free encyclopedia

Inverse distance weighting (IDW) is a method for multivariate interpolation, a process of assigning values to unknown points by using values from usually scattered set of known points.

A general form of finding an interpolated value u for a given point x using IDW is an interpolating function:

$u(\mathbf{x}) = \frac{ \sum_{k = 0}^{N}{ w_k(\mathbf{x}) u_k } }{ \sum_{k = 0}^{N}{ w_k(\mathbf{x}) } },$

where:

$w_k(\mathbf{x}) = \frac{1}{d(\mathbf{x},\mathbf{x}_k)^p},$

is a simple IDW weighting function, as defined by Shepard^[1], x denotes an interpolated (arbitrary) point, x_k is an interpolating (known) point, $d$ is a given distance (metric operator) from the known point x_k to the unknown point x, N is the total number of known points used in interpolation and $p$ is a positive real number, called the power parameter. Here weight decreases as distance increases from the interpolated points. Greater values of $p$ assign greater influence to values closest to the interpolated point. For 0 < p < 1 u(x) has sharp peaks over the interpolated points x_k, while for p > 1 the peaks are smooth. The most common value of $p$ is 2.

The Shepard's method is a consequence of minimization of a functional related to a measure of deviations between tuples of interpolating points {x, u} and k tuples of interpolated points {x_k, u_k}, defined as:

$\phi(\mathbf{x}, u) = \left( \sum_{k = 0}^{N}{\frac{(u-u_k)^2}{d(\mathbf{x},\mathbf{x}_k)^p}} \right)^{\frac{1}{p}} ,$

derived from the minimizing condition:

$\frac{\part \phi(\mathbf{x}, u)}{\part u} = 0.$

The method can easily be extended to higher dimensional space and it is in fact a generalization of Lagrange approximation into a multidimensional spaces. A modified version of the algorithm designed for trivariate interpolation was developed by Robert J. Renka and is available in Netlib as algorithm 661 in the toms library.

1 Liszka's method
2 Probability metric
3 References
4 See also

[edit] Liszka's method

A modification of the Shepard's method was proposed by Liszka^[2] in applications to experimental mechanics, who proposed to use:

$w_k(\mathbf{x}) = \frac{1}{(d(\mathbf{x},\mathbf{x}_k)^2+ \varepsilon^2)^\frac{1}{2}},$

as a weighting function, where ε is chosen in dependence of the statistical error of measurement of the interpolated points.

[edit] Probability metric

Yet another modification of the Shepard's method was proposed by Łukaszyk^[3] also in applications to experimental mechanics. The proposed weighting function had the form:

$w_k(\mathbf{x}) = \frac{1}{(D_{**}(\mathbf{x}, \mathbf{x}_k) )^\frac{1}{2}},$

where $D_{**}(\mathbf{x}, \mathbf{x}_k)$ is a probability metric chosen also with regard to the statistical error probability distributions of measurement of the interpolated points.